I've put together an example air application using longurlplease, based on what @Swizec sent me, but without jquery, so that it will make sense to you no matter what other js libraries you may be using.
I use the URLLoader class to make calls to the api and this JSON parser to parse the responses.
Thanks @Swizec for helping me figure this out. Hopefully it will help more people lengthen short urls in their adobe air applications.